Mean, Median, Mode, and Range

The mean, median, unique mode, and range of a collection of eight counting numbers are each 8. What is the largest number that can be in this collection?
Source: mathcontest.olemiss.edu 10/23/2006

SOLUTION
First solution
Let’s try a simple solution and see what we got
1\;\;2\;\;3\;\;8\;\;8\;\;8\;\;8\;\;9
Median = 8
Mode = 8
Range = 9-1=8
Mean = \frac{1+2+3+8+8+8+8+9}{8}=5.875\neq 8
We got everything right except the mean.

Second solution
4\;\;8\;\;8\;\;8\;\;8\;\;8\;\;8\;\;12
Median = 8
Mode = 8
Range = 12-4=8
Mean = \frac{4+8+8+8+8+8+8+12}{8}=8
We got everything right, but is 12 the largest value?

Third solution
5\;\;6\;\;8\;\;8\;\;8\;\;8\;\;8\;\;13
Median = 8
Mode = 8
Range = 13-5=8

Mean = \frac{5+6+8+8+8+8+8+13}{8}=8
We got everything right, but is 13 the largest value?

Fourth solution
6\;\;7\;\;7\;\;8\;\;8\;\;8\;\;8\;\;14
Median = 8
Mode = 8
Range = 14-6=8
Mean = \frac{6+7+7+8+8+8+8+14}{8}=8.25\neq 8
We got 14 as the largest value, but the mean is a little off.

Fifth solution
6\;\;6\;\;6\;\;8\;\;8\;\;8\;\;8\;\;14
Median = 8
Mode = 8
Range = 14-6=8
Mean = \frac{6+6+6+8+8+8+8+14}{8}=8
We got everything right, but is it the end?

7\;\;7\;\;7\;\;8\;\;8\;\;8\;\;8\;\;15
Median = 8
Mode = 8
Range = 15-7=8
Mean = \frac{7+7+7+8+8+8+8+15}{8}=8.5\neq 8
So the largest value in the collection is 14.

Answer: 14.

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About mvtrinh

Retired high school math teacher.
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